The stream function, Airy-stress function formulation of the classical Stokes problem in the plane has not been exhaustively studied as an elliptic boundary value problem. Bitsadze appears to have been the first to question the well-posedness of this formulation subject to certain boundary conditions. The div-curl formulation [Tahir, 1999b] has enabled us to present existence and uniqueness results for the solution of Stoke-Bitsadze Problem. Stoke-Bitsadze Problem-I corresponds to the boundary conditions prescribed by Vanmaele et al [1994].